In equation (28) of theorem 3.6.1, it says that "where t0 is any conveniently chosen point in I".

This is for the integral limits for u1 and u2.

I don't understand how we can just conveniently choose some t0 for the general solution when it's not an initial value problem.

Looking for a general solution we can select initial point as we please; it definitely affects other constants. Look at

\begin{equation*}

\int_{t_0}^f f(t)\,dt'+C_0=\int_{t_1}^f f(t)\,dt'+C_1

\end{equation*}

as long as $C_1-C_0=-\int_{t_0}^{t_1}f(t')\,dt'$ and you can select any initial point $t_0$.